Question: Solve for $x$ and $y$ using elimination. $\begin{align*}2x-2y &= 2 \\ 3x+4y &= -4\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $2$ and the bottom equation by $1$ $\begin{align*}4x-4y &= 4\\ 3x+4y &= -4\end{align*}$ Add the top and bottom equations. $7x = 0$ Divide both sides by $7$ and reduce as necessary. $x = 0$ Substitute $0$ for $x$ in the top equation. $2( 0)-2y = 2$ $-2y = 2$ $-2y = 2$ $y = -1$ The solution is $\enspace x = 0, \enspace y = -1$.